The idea of parallelism

In that formulation, the connection is encoded at each point p∈Pp \in P in the total space by a decomposition of the tangent spaceTpPT_p P as a direct sumTpP≃Vp⊕HpT_p P \simeq V_p \oplus H_p of vector spaces, such that

Vp=kerπ*|pV_p = \ker \pi_*|_p is the kernel of the projection map that sends vectors in the total space to vectors in base space (this part is fixed by the choice of p:P→Xp : P \to X);

Hp⊂TpPH_p \subset T_p P is a choice of complement, such that this choice varies smoothly over PP in an evident sense and is compatible with the GG-action on PP.

The vectors in VpV_p are called vertical , the vectors in HpH_p are called horizontal . One may think of this as defining locally in which way the base space sits horizontally in the total space, equivalently as identifying locally a “smoothly varying local trivialization” of PP.

More precisely, given such a choice of horizontal subspaces, there is for every path γ:[0,1]→X\gamma : [0,1] \to X and every choice of lift γ^(0)∈P\hat \gamma(0) \in P of the start point γ(0)\gamma(0) to the total space of the bundle, a unique liftγ^:[0,1]→P\hat \gamma : [0,1] \to P of the entire path to the total space:

In other words, this means that given a path γ\gamma down in XX, we may transport any point p∈Pγ(0)p \in P_{\gamma(0)} above its start point parallely (with respect to the notion of parallelism determined by ∇\nabla) along γ\gamma, to find a uniquely determined point tra∇(γ)(p)∈Pγ(1)tra_\nabla(\gamma)(p) \in P_{\gamma(1)} over the endpoint.

The smooth paths in a smooth manifold XX naturally form the diffeological groupoid called the path groupoidP1(X)P_1(X). Objects are points in XX, morphsims are thin homotopy-classes of smooth paths which are constant in a neighbourhood of their boundary, composition is concatenation of paths.

For P→XP \to X any GG-bundle, there is also naturally the diffeological groupoid At(P)At(P) – the Atiyah Lie groupoid of PP. Objects are points in XX, morphisms are homomorphisms of GG-torsors between the fibers over these points.

Then the above properties of parallel transport are equivalent to saying that we have an internal functor

tra:P1(X)→At(P)
tra : P_1(X) \to At(P)

that is the identity on objects. Moreover, this functor uniquely characterizes the connection on PP that it comes from. This means that we may identify connections on PP with their parallel transport functors.

But even the bundle PP itself is encoded in such functors. If instead of looking at the category of internal groupoids and internal functors, we look at the larger 2-topos of diffeological stacks – stacks over CartSp.

Then we can take simply the diffeological delooping groupoid BG\mathbf{B}G, which has a single object and GG as its hom-set and consider morphisms

tra:P1(X)→BG
tra : P_1(X) \to \mathbf{B}G

in the 2-topos. These are now given by anafunctors of internal groupoids, and one finds that they encode a Cech cocycle for a GG-principal bundle PP together with the parallel transport of a connection over it.

There is also the diffeological groupoid incarnation of the fundamental groupoidΠ1(X)\Pi_1(X) of XX. Its morphisms are full homotopy-classes of paths. There is a canonical projection P1(X)→Π1(X)P_1(X) \to \Pi_1(X) that sends a thin-homotopy class of paths to the corresponding full-homotopy class.

A parallel transport functor tra:P1(X)→Gtra : P_1(X) \to G factors through Π1(X)\Pi_1(X) precisely if the corresponding conneciton is flat in that its curvature form vanishes.

In physics

The forces exerted by such gauge fields on charged particles propagating on XX (i.e. electrons, quarks and generally massive particles, respectively) are encoded precisely in the parallel transport assignment of the gauge field connection to their trajectories.

More precisely, the exponentiated action functional for the electron propagating on XX in the presence of an electromagnetic field ∇\nabla is the functional on the space of paths in XX given by

where the first term is the standard kinetic action. If ∇\nabla is a (nontrivial) connection on a trivial bundle, then, as described below it is encoded by a differential formA∈Ω1(X)A \in \Omega^1(X) – called the vector potential in physics – and we have

If instead of looking at the quantum mechanics of the quantum particle charged under a fixed background gauge field look at the quantum field theory of that gauge field itself, we can use the action functional of particles to probe these background fields and obtain quantum observables for them.

This converse assignment where we fix a path γ\gamma and regard the parallel transport then as a functional over the space of all connections over XX

For γ:[0,1]→X\gamma : [0,1] \to X we have the pull-back 1-form γ*A∈Ω1([0,1])\gamma^* A \in \Omega^1([0,1]). For f∈C∞([0,1],G)f \in C^\infty([0,1], G) a smooth function with values in the Lie groupGG, consider the differential equation

df+ρ(f)*(γ*A)=0,
d f + \rho(f)_*(\gamma^*A) = 0
\,,

where df:T[0,1]→TGd f : T [0,1] \to T G is the differential of ff and where ρ:G×G→G\rho : G \times G \to G is the left action of GG on itself (i.e. just the multiplication on GG) and r(f)*:TG→TGr(f)_* : T G \to T G its differential and using the defining identification 𝔤≃TeG\mathfrak{g} \simeq T_e G we take r(f)*(A)r(f)_*(A) to be the composite T[0,1]→γ*A𝔤↪TG→r(f)*TGT [0,1] \stackrel{\gamma^* A}{\to} \mathfrak{g} \hookrightarrow T G \stackrel{r(f)_*}{\to} T G.

is the value f(1)∈Gf(1) \in G for the unique solution of the equation df+ρ(f)*(A)=0d f + \rho(f)_*(A) = 0 with initial value f(0)=ef(0) = e (the neutral element in GG).

The notation here is motivated from the special case where G=ℝG = \mathbb{R} is the group of real numbers. In that case the Lie algebra𝔤≃ℝ\mathfrak{g} \simeq \mathbb{R} is abelian, the differential equation above is simply

df=γ*(A)∧f
d f = \gamma^*(A) \wedge f

for a real valued function f∈C∞([0,1])f \in C^\infty([0,1]), and the unique solution to that with f(0)=e=0f(0) = e = 0 is literally the exponential of the integral of AA: